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Clifford (i.e. "geometric") algebras are of some mathematical interest, but for physics the only real value I've seen from them is in offering a fairly nice pre
by nyssos 3y ago
Clifford (i.e. "geometric") algebras are of some mathematical interest, but for physics the only real value I've seen from them is in offering a fairly nice presentation of spin groups, which is ultimately where the Dirac matrices come from. The geometric algebra advocacy, so far as I can tell, comes entirely from engineers who were never given a proper account of tensors in the first place: it's certainly an improvement over whatever godawful basis-dependent stuff gets taught there.
- itishappy 3y agoI'd be interested to know your opinion on space-time algebras. They seem like they would provide a nice way of unifying spatial rotations and Lorentz boosts, but that may be blind optimism as your last sentence seems to have been written to describe me specifically...
- nyssos 3y agoSpacetime algebra is just the Clifford algebra Cl_1_3(R), so all of the above applies. > They seem like they would provide a nice way of unifying spatial rotations and Lorentz boosts Yes and no: the right way to unify rotations and boosts is to consider them as the orientation-preserving elements of the Lorentz group (sometimes called the proper Lorentz group). You can construct this from the corresponding Clifford algebra, but it's somewhat technical and not physically well-motivated until you start dealing with spinors. It's also the group of symmetries of spacetime that leave the origin unchanged, which is far, far more natural.
- paholg 3y agoI found it a much nicer way to think about physical concepts. Torque and magnetism, for example, make much more sense as bivectors. And never having to do the right-hand rule is a nice plus.